Question

In particle physics planck length ($l_p$) is calculated in terms of three physical constants: the speed of light (C), Planck constant (h) and universal gravitational constant (G). The expression for $l_p$ can be:

Answer 1

$l_p = \sqrt{\frac{G\,h}{c^3}}$

Answer 2

We assume the Planck length can be written as

$l_p = G^a \, h^b \, c^d$.

The dimensions of the constants are:

$[G] = L^3 M^{-1} T^{-2},\quad [h] = M\, L^2\, T^{-1},\quad [c] = L\, T^{-1}$.

Thus, the dimensions of $l_p$ become:

$l_p = L^{3a}M^{-a}T^{-2a}\cdot L^{2b}M^bT^{-b}\cdot L^dT^{-d} = L^{3a+2b+d} \, M^{-a+b} \, T^{-2a-b-d}$.

For $l_p$ (a length) the overall dimensions must be $L^1$. That gives:

1. Mass: $-a + b = 0$ ⟹ $b = a$.
2. Length: $3a + 2a + d = 5a + d = 1$ ⟹ $d = 1 - 5a$.
3. Time: $-2a - b - d = -3a - d = 0$ ⟹ $d = -3a$.

Equate the two expressions for $d$:

$1 - 5a = -3a \quad\Rightarrow\quad 1 = 2a \quad\Rightarrow\quad a = \frac{1}{2}$.

Then,

$b = \frac{1}{2} \quad \text{and} \quad d = -\frac{3}{2}$.

Thus,

$l_p = G^{1/2}\,h^{1/2}\,c^{-3/2} = \sqrt{\frac{G\,h}{c^3}}$.